denoising of tdm signal using novel time domain and transform domain multirate adaptive algorithms

American Journal of Engineering Research (AJER) 2014

w w w . a j e r . o r g

Page 212

American Journal of Engineering Research (AJER)

e-ISSN : 2320-0847 p-ISSN : 2320-0936

Volume-03, Issue-08, pp-212-226

www.ajer.org

Research Paper Open Access

Denoising Of TDM Signal Using Novel Time Domain and

Transform Domain Multirate Adaptive Algorithms

C Mohan Rao1, Dr. B Stephen Charles

2, Dr. M N Giri Prasad

3

1Asso. Prof., Department of ECE, NBKRIST, Vidhyanagar, India 2Principal, SSCET, Kurnool, India

3Prof. & Head, Department of ECE, JNTUACE, Anantapuram, India

ABSTRACT : The Gaussian noise is obvious in most of the communication channels. The impulsive noise

tends to Gaussian form as the time slot extends over a significant period. In this paper a channel with Gaussian

noise is considered. A TDM signal passed through this kind of channel was applied to denoising. The denoising

was implemented in time as well as transform domain. The adaptive algorithms like LMS, NLMS, RLS, LSL and

transform domain LMS implementations using wavelet packets are designed. Two new algorithms are proposed

and verified with the above application; one a variation of LMS and second binary step size LMS. From the

simulation results it was found that the multirate adaptive algorithms proposed give better performance than the

existing techniques.

KEYWORDS :Denoising, TDM, LMS, Wavelet Packet

I. LMS ALGORITHM

The simplified block diagram of a transversal adaptive FIR filter is depicted in figure 1, where the

block denoted by adaptive filter comprises as adaptive filter t

Nnhnhnhnh )(ˆ),.......,(ˆ),(ˆ)(ˆ

21 and algorithm, x(n) is

the input sequence from which the input vector t

NnxnxnxnX )1(),......,1(),()( is obtained, e(n) is the

output error, )(ˆ ny is the output of the adaptive filter and d(n) is the desired signal. All the theoretical

derivations are referred to figure 1.

Fig. 1 Simplified block diagram of an adaptive FIR filter

In connection with figure 1, the output of the adaptive filter can be written as follows:

N

i

i

ttinxnhnhnXnXnhny

1

)1()(ˆ)(ˆ)()()(ˆ)(ˆ (1)

wheret is the transposition operator. The output error is expressed by the following equation [1][2]:



13

)(ˆ)()( nyndne (2)

The coefficients of the adaptive filter are updated to minimize the output mean squared error defined as

follows:

22)](ˆ)([)()( nyndEneEnJ (3)

The optimum filter coefficients in the mean square sense are those coefficients for which the partial

derivatives of J(n) equals to zero. Denoting the vector of the optimum coefficients as t

oNoohhh ,....,

1 , the

system of equations which gives ho is obtained as in the sequel:

NneinxE

nyndinxE

h

nyndE

h

nJ

io

o

oioi

,...,.1,0)}()1({2

)]}()()[1({2

)]()([)(2

(4)

where )()()( nXhndnet

oo (5)

is the minimum output error obtained when the coefficients of the adaptive filter equals the coefficients

of the optimum Wiener filter. Equation (4) can be written in a more compact form as follows:

0)]()([ nenXEo

(6)

It follows from (6) that the optimum error is orthogonal to the input vector at each time instant n, and

this represents the well-known principle of orthogonality. From equation (4), the Wiener-Hopf equations which

give the coefficients of the optimum filter are represented by:

pRh

NpNNrhNrhNrh

pNrhrhrh

pNrhrhrh

o

oNoo

oNoo

oNoo

)()(.....)2()1(

.............................................................................

)1()2(.....)22()12(

)0()1(.....)21()11(

21

21

21

(7)

where )]()([)( jnxinxEjir ,

)]()([)( inxndEip

andt

Npppp )](),...2(),1([

We note that the terms )()( ijrjir and jirjjriir ,)0()()( , therefore the

matrix R can be written as:

)0(.....)3()2()1(

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

.

)2(.....)1()0()1(

)1(.....)2()1()0(

rNrNrNr

Nrrrr

Nrrrr

R (8)

When the matrix R is invertible and its elements can be estimated, the optimum Wiener filter can be

easily obtained from (7) as:

pho

1 R (9)

In situations when the elements of the matrix R are not available an iterative algorithm can be applied

to the adaptive filter which transforms its coefficients toward ho. One simple adaptive algorithm is the Steepest

Descent (SD) algorithm, which updates the coefficients of the adaptive filter at each iteration in the opposite

direction of the cost function gradient. In the case of the SD, the update formula for the filter coefficients is:



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)(2

1)(ˆ)1(ˆ nJnhnh (10)

where

t

Nnh

nJ

nh

nJ

nh

nJnJ

)(ˆ

)(,.......,

)(ˆ

)(,

)(ˆ

)()(

21

and

)]()1([2

)(ˆ

)(neinxE

nh

nJ

i

(11)

In order to compute the elements of the gradient in equation (11), the expectation operator must be

used. A simpler alternative is to use the instantaneous gradient instead of the true gradient and the obtained

algorithm is called the Least Mean Square (LMS). As a consequence, the LMS algorithm uses the following

coefficient update formula:

)()()(ˆ)1(ˆ nXnenhnh (12)

where the step-size µ was introduced to control the stability of the algorithm. Finally, the LMS

algorithm can be described by the following four steps:

1. From the input vector t

NnxnxnxnX )]1(,),........1(),([)( from the input sequence x(n).

2. Compute the output of the adaptive filter: )()(ˆ)(ˆ)()(ˆ nXnhnhnXnytt

.

3. Compute the output error: )(ˆ)()( nyndne .

4. Update the coefficients of the adaptive filter: )()()(ˆ)1(ˆ nXnenhnh .

II. VARIATION OF LMS If it were possible to make exact measurements of the gradient vector in all iterations and if the step-

size parameter is suitably chosen, then the tap-weight vector computed by using the method of steepest-

descent would indeed converge to the optimum Wiener solution. In reality, however, exact measurements of the

gradient vector are not possible, and it must be estimated from the available data. In other words, the tap-weight

vector is updated in accordance with an algorithm that adapts to the incoming data. One such algorithm is the

least mean square algorithm. A significant feature of LMS is its simplicity; it does not require measurements of

the pertinent correlation functions, nor does it require matrix inversion. The LMS algorithm is a search

algorithm in which a simplification of the gradient vector computation is made possible by appropriately

modifying the objective function. The LMS algorithm, as well as others related to it, is widely used in various

applications of adaptive filtering due to its computational simplicity. The convergence speed of the LMS is

shown to be dependent on the eigenvalue spread of the input signal correlation matrix. The LMS algorithm is by far the most widely used algorithm in adaptive filtering for several reasons. The main features that attracted the

use of the LMS algorithm are low computational complexity, proof of convergence in stationary environment,

unbiased convergence in the mean to the Wiener solution, and stable behavior when implemented with finite-

precision arithmetic. Let x(n) and d(n) represent the reference input and the desired output signal, respectively,

to the adaptive filter. Let L denote the total number of filter coefficients. Define the L x 1 coefficient vector H(n)

and the input vector X(n) as

H(n) = [ho(n),h1(n),...,hL-1(n)]T (13)

X(n)=[x(n),x(n-l),…,x(n- L+l)]T (14)

The LMS is described as

e(n) = d(n)- HT(n)X(n) (15)

H(n + 1) = H(n) + µsX(n)e(n) (16)

In practice, (16) may be replaced with

)()()()(

)()1( nenXnXnX

nHnHT

(17)

or

)()()0(

)()1( nenXLr

nHnH

(18)

where the positive step-size µ is bounded by 2, σ is a small positive number and r(0)is the estimated



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autocorrelation function value of x(n) for lag 0.

Digital filter is the basic building block of Digital Signal Processing systems. Finite Impulse Response

is preferred when compared to Infinite Impulse Response, because of its properties like guaranteed stability,

linear phase and low response, but with expense of large number of arithmetic operations are involved. In

communication systems channel noise and Inter Symbol Interference degrades the performance of

communication system. To reduce this problem adaptive equalizers are used to shape the signals at the receivers. Least Mean Square technique is the one of the adaptive techniques. It is easy to realize, the

computational complexity causes a long output delay, which is not tolerable. This computational complexity can

be reduced using frequency domain adaptive filtering, but nonlinear systems performance degrades drastically.

To overcome this, problem of derivative base and derivative free learning algorithm, we use natural selection or

derivative free algorithms. A new evolutionary computation algorithm based on natural learning to update the

weights of adaptive filter was used. In this method Genetic Algorithm (GA) is used to update weights of filter

coefficients.

III. BINARY STEP-SIZE LMS In practical applications adaptive algorithms which possess high convergence speed while maintaining

small convergence error rate are of great interest. For instance, in channel equalization during the transient

period, the frequency characteristic of the adaptive equalizer is far from the inverse of the frequency response of

the channel therefore the data transmitted during this time will be corrupted. In echo cancellation application, if

the coefficients of the adaptive canceler are not close to the coefficients of the FIR filter which models the echo

path the resulting echo signal is not attenuated. Actually, it is possible in this application, that during the

transient period, the echo will be actually amplified. As a consequence, the transient period of the adaptive filter

must be as small as possible for most of the practical applications in order to improve the overall quality of the

system. The LMS algorithm has a small computational complexity therefore; it is very simple to be

implemented in practice. Although it is simplicity, one of its main drawbacks is the fact that the speed of

convergence and steady state error depends on the same parameter, the step-size µ.

In conclusion, when a constant step-size is used in LMS, there is a tradeoff between the steady-state

error and the convergence speed, which prevent a fast convergence when the step-size is chosen to be small for

small output error. In order to deal with this problem, a simple idea is to use a step-size which is time-varying

during the adaption. At early stages of the adaption, when the adaptive filter is far from the optimum, a larger

value of the step-size should be used. This will shorten the transient period and increase the convergence speed

of the adaptive filter. As the adaptive filter goes close to the optimum Wiener solution, the step-size should be

decreased and so the misadjustment. The adaptive algorithms derived from the LMS, which uses time-varying

step-size modified as described above, belong to the class of variable step-size LMS algorithms. The variable

step-size LMS algorithm first introduced by Kwong and Johnston in [4] uses the following update formula for

the adaptive filter coefficients:

)()()()(ˆ)1(ˆ nxnennhnh (19)

where )(ˆ nh is the N x 1 vector of the adaptive filter coefficients, x(n) is the vector of the past N samples from

the input sequence, µ(n) is a time-varying step-size and e(n) is the output error.The time-varying step-size is

also adapted as in the following equation:

)()(')1('2

nenn ,

otherwisen

nif

nif

n

)1('

)1('

)1('

)1(minmin

maxmax

(20)

with 0 < α < 1 and γ > 0 being some constant parameters and µmax and µmin being the upper and lower bounds of

the time-varying step-size. The constant parameter µmax which is normally selected close to the instability point

of the conventional LMS algorithm is used to increase the convergence speed, while the parameter µmin is

chosen provide a good compromise between the steady-state misadjustment and the tracking capacity of the

algorithm. The parameter γ is used to control the convergence time and also the steady-state level of the



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misadjustment. The behavior of the step-size as described in (20) is the following: at early stages of the adaption

the step-size is increased due to the large value of the output error. As the algorithm goes closer to the steady-

state the value of e(n) decreases which decrease the step-size µn.The following approximate analytical

expression for the steady-state misadjustment of the variable step-size LMS algorithm was derived in [4]:

][1

)3(211

][1

)3(211

2

min

2

min

min

R

R

trJ

trJ

J

JM

ex

(21)

Clearly, the steady-state misadjustment depends on the parameter γ and on the minimum value of the

MSE Jmin. Since the speed of convergence of the algorithm depends also on the parameter γ, one can conclude

that there is still dependence between the misadjustment and the convergence time [5]. Another drawback of this

algorithm is the fact that the steady-state misadjustment depends also on Jmin. For instance in system

identification applications, the minimum MSE equals the output noise variance, therefore the steady-state

misadjustment depends on the system noise.In this section, a new variable step-size algorithm is presented in

which the step-size varies in two ways thereby naming binary step-size variation algorithm. Supervised channel

equalization is considered [6]. Depending on the error the step-size gets updated. The updation process is shown

in the figure 2.

Fig. 2Updation process

In another observation it has been identified that if with the NLMS the maximum change of step-size is

limited to 0.01, the convergence speed is very high [7].

IV. TRANSFORM DOMAIN LMS The transform-domain LMS algorithm is another technique to increase the convergence speed of the

LMS algorithm when the input signal is highly correlated. The basic idea behind this methodology is to modify

the input signal to be applied to the adaptive filter such that the conditioning number of the corresponding

correlation matrix is improved. In the transform-domain LMS algorithm, the input signal vector x(k) is transformed in a more convenient vector s(k), by applying an orthonormal (or unitary) transform [4], i.e.,

s(k) = Tx(k) (22)

whereTTT= I. The MSE surface related to the direct-form implementation of the FIR adaptive filter can be

described byξ(k) = ξmin +ΔwT(k)RΔw(k) (23)

whereΔw(k) = w(k) − wo. In the transform-domain case, the MSE surface becomes

)(ˆ)(ˆ

)(ˆ)]()([)(ˆ)(

min

min

kwkw

kwksksEkwk

TT

TT

TRT

(24)

where )(ˆ kw represents the adaptive coefficients of the transform-domain filter. Figure 3 depicts the transform-

domain adaptive filter. It can be noticed that the eccentricity of the MSE surface remains unchanged by the

application of the transformation, and, therefore, the eigenvalue spread is unaffected by the transformation [8].

As a consequence, no improvement in the convergence rate is expected to occur. The Transform-Domain LMS

Algorithm is as follows:

Initialization T

wx ]0...00[)0(ˆ)0(

γ = small constant

0 < α ≤ 0.1

Do for each x(k) and d(k) given for k≥0

s(k) = Tx(k)



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)(ˆ)()()( kwkskdke

T

)()()(2)(ˆ)1(ˆ

2kskkekwkw

In the literature, Karhunen-Lo`eve Transform (KLT) is used as unitary transform for the transform-

domain adaptive filter. However, since the KLT is a function of the input signal, it cannot be efficiently

computed in real time. An alternative is to choose a unitary transform that is close to the KLT of the particular

input signal [9]. By close is meant that both transforms perform nearly the same rotation of the MSE surface. In

any situation, the choice of an appropriate transform is not an easy task. Some guidelines can be given, such as

(however these are just conventions not rules):

1. Since the KLT of a real signal is real, the chosen transform should be real for real input signals;

2. For speech signals the discrete-time cosine transform (DCT) is a good approximation for the KLT;

3. Transforms with fast algorithms should be given special attention. A number of real transforms such as

DCT, discrete-time Hartley transform, and others, are available.

Most of them have fast algorithms or can be implemented in recursive frequency-domain format [10]. In particular, the outputs of the DCT are given by

N

i

olkx

N

ks

0

)(

1

1)( (25)

N

l

i

N

lilkx

Nks

0 )1(2

)12(cos)(

1

2)( (26)

For complex input signals, the discrete-time Fourier transform (DFT) is a natural choice due to its

efficient implementations [11]-[14].

Fig. 3 Transform-domain adaptive filter

V. WAVELET PACKET TRANSFORM In this paper, the unitary transform used is wavelet packet transform. For almost all signals, the low-

frequency component is the most important part. It is what gives the signal its significance and identity. The

high-frequency content, on the other hand, adds flavor. Consider an audio signal.



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If the high frequency components are removed, the audio sounds different, but one can still tell what's

being said in the audio. However, if enough of the low-frequency components are removed, one hears

gibberish.Wavelet analysis often speaks about approximations and details. The approximations are the low-

frequency, high-scale components of the signal. The details are the high-frequency, low-scale components. The

filtering process in wavelet analysis, at its basic level, looks something like figure 4.The original sequence, S,

applied to two complementary filters and emerges as two signals as shown in figure 4. If a digital sequence of

say 512 samples is applied to the filter bank consisting of one low and one high pass filter as mentioned above, the length of A will be 512 and that of D will also be 512. Hence the data to handle was doubled. But note that

in A as well as in D only 256 samples are irredundant. To remove the redundant samples, the downsamplers are

employed as shown in figure 5. The outputs are denoted by cA and cD.

Fig. 4 Filtering Process in Wavelet Analysis

Fig. 5 Wavelet processing with downsamplers

This process, i.e., the conversion of S into cA and cD is called decomposition; the filters at this stage

are referred as decomposition low pass and decomposition high pass filters. These filters have direct relation to

the basis function used in a specific wavelet. The vectors cA and cD constitutes the DWT coefficients.

a. Multiple Stages of Decomposition

The decomposition process can be repeated means iterated, with successive approximations being

decomposed in turn, so that one signal is broken down into many lower resolution components. This is called

the wavelet decomposition tree shown in figure 6.

Fig. 6 Multistage Decomposition

The maximum number of decomposition stages should be taken so that the length of the sequence in

the last stage is not less than 1. From the wavelet coefficients the original signal need to be recovered. The process of obtaining the original signal by using the wavelet coefficients is called reconstruction or synthesis



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shown in figure 7.The downsampling performed at decomposition stage introduces an aliasing effect. The

reconstruction filters need to be selected so that the aliasing effect introduced at the decomposition stage should

be cancelled. The overall process of wavelet is depicted in the figure 8.The wavelet packet analysis is an

extension of wavelet analysis with an inclusion of analysis of both approximation (cA) and detail (cD)

components. The wavelet packet analysis looks like a complete tree structure. The multistage wavelet packet

analysis looks like as shown in figure 9.

Fig. 7 Reconstruction Stage

Fig. 8 Wavelet Transform as a mulirate filter

The wavelet packet analysis is an extension of wavelet analysis with an inclusion of analysis of both

approximation (cA) and detail (cD) components. The wavelet packet analysis looks like a complete tree

structure. The multistage wavelet packet analysis looks like as shown in figure 9.

Fig. 9 Wavelet Packet Analysis

The wavelet packets use the wavelet filters to decompose and reconstruct the signals. The wavelet filters corresponds to the perfect reconstruction condition as well as to represent the data to suite different

applications.



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VI. SIMULATION RESULTS In this section the simulation results of adaptive filters, Wavelets Packet based transform domain LMS,

the variation of LMS (Var LMS), Binary step size LMS (BSS LMS), and multirateVar LMS and multirate BSS

LMS in wavelet packet domain on de-noising of TDM signal are presented. The flow graph in figure 10

represents the total process that was considered in this paper.

Fig. 10 De-noising of TDM signals based on Wavelet Packet based denoising

The two signals that are multiplexed in TDM, multiplexed signal and noisy signals are shown in the figure 11.

Fig. 11 Input, TDM and Noisy signals

The Peak Signal to Noise Ratio (PSNR), Mean Square Error (MSE), Maximum Squared Error (MAX ERR), Ratio of Squared Norms (L2RAT) are calculated and tabulated for different types of techniques.The

simulation results of adaptive filters on denoising are presented in table I. LMS, Normalized LMS, RLS and

LSL based denoising are implemented. It is observed that the maximum value of maximum error is 5.03 with

LMS in the second signal case. The standard LMS has produced highest MSE and LSL a lowest MSE.



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Table I: Denoising with Adaptive Filters

Denoising with Adaptive Filters

PSNR MSE MAX ERR L2RAT

FS SS FS SS FS SS FS SS

LMS 49.19 48.02 0.78 1.02 4.21 5.03 0.93 0.98

NLMS 52.30 52.32 0.38 0.38 2.04 3.17 0.95 0.93

RLS 53.37 52.39 0.30 0.38 1.93 2.74 0.98 1.03

LSL 54.02 53.87 0.26 0.27 1.56 2.13 0.91 0.96

Average 52.22 51.65 0.43 0.51 2.44 3.27 0.95 0.97

Table II presents the simulation results of denoising using LMS based on DCT and DFT. The

performance of these techniques is almost similar to that of the previous adaptive filters except LMS. The standard LMS has shown fragile performance.

Table II: Denoising with Transform domain LMS based on DCT & DFT

Denoising with Transform domain LMS based on DCT & DFT



DFT 53.30 53.04 0.30 0.32 1.81 2.14 1.02 0.93

DCT 53.88 51.62 0.27 0.45 2.08 4.15 0.92 0.88

Average 53.59 52.33 0.28 0.39 1.94 3.15 0.97 0.91

Table III shows the performance of LMS in wavelet packet domain with different wavelets. Haar,

Daubechies, Symlets, Coiflets, Biorthogonal, demeyer and reverse biorthogonal wavelets are considered. The

performance with different wavelet combinations is almost similar. The PSNR is in the range of 54dB with

almost all wavelets. The maximum error is around 1.6.

Table III: Denoising with LMS in Wavelet Packet Domain

Denoising with LMS in Wavelet Packet Domain



Haar 54.02 54.12 0.26 0.25 1.78 1.87 1.08 1.13

db10 54.51 53.76 0.23 0.27 1.33 1.86 1.06 1.06

db45 54.64 54.15 0.22 0.25 1.39 1.54 1.04 1.06

sym6 54.41 54.09 0.24 0.25 1.25 1.55 1.04 1.02

coif4 54.89 53.56 0.21 0.29 1.56 1.67 1.09 1.06

bior2.4 54.29 54.06 0.24 0.26 1.52 1.79 1.02 1.08

dmey 54.01 54.11 0.26 0.25 1.62 1.76 1.02 1.10

rbio1.3 54.39 54.05 0.24 0.26 1.70 1.78 1.04 1.06

Average 54.40 53.99 0.24 0.26 1.52 1.73 1.05 1.07

Table IV gives the simulation results of variation of LMS on denoising in five runs. The performance is

almost similar to that of the previous techniques. The maximum value of PSNR is 54.08dB while the maximum

error is 4.37.



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Table IV: Denoising with Var LMS

Denoising with Var LMS



Var LMS-1 53.34 51.14 0.30 0.50 3.53 3.67 0.91 0.88

Var LMS-2 54.08 52.82 0.25 0.34 2.93 4.09 0.92 0.89

Var LMS-3 53.15 51.79 0.31 0.43 2.87 4.37 0.94 0.83

Var LMS-4 53.07 51.54 0.32 0.46 3.30 4.11 0.96 0.82

Var LMS-5 53.84 51.95 0.27 0.42 2.77 3.91 0.86 0.94

Average 53.50 51.85 0.29 0.43 3.08 4.03 0.92 0.87

Table V presents the performance of BSS LMS on denoising. The table gives the values of parameters

in five runs. The average value of PSNR wit first signal is 53.07dB, with second signal 53.99dB, while the

maximum error is 1.52 and 2.73 with two signals respectively. The performance of BSS LMS is found to be

slightly better than the traditional LMS and transform domain LMS based on DCT and DFT and the variation of LMS, but the improvement is not significant.

Table V: Denoising with BSS LMS

Denoising with BSS LMS



BSS LMS-1 51.83 51.93 0.43 0.42 4.56 4.81 0.87 0.97

BSS LMS-2 54.22 52.11 0.25 0.40 2.52 3.70 0.89 0.89

BSS LMS-3 52.83 51.38 0.34 0.47 2.58 4.02 0.95 0.84

BSS LMS-4 53.33 52.11 0.30 0.40 2.99 3.92 0.91 0.86

BSS LMS-5 53.18 52.37 0.31 0.38 2.76 3.16 0.94 0.85

Average 53.07 51.98 0.33 0.41 3.08 3.92 0.91 0.88

Table VI presents the performance of the multirate variation of LMS by using wavelet packet domain

on denoising. The simulation results show that the performance is improved by a great extent. The PSNR values

at times 66.37dB and 64.46dB. The minimum value of PSNR is 58.86 with demeyer wavelet. The maximum

error is in the range of 1.5. The results are obvious because of the correlated data coming out of wavelet packet

representation.

Table VI: Denoising with MultirateVar LMS using Wavelet Packet Domain

Denoising with MultirateVar LMS using Wavelet Packet Domain



Haar 60.10 66.37 0.06 0.02 1.50 1.31 1.04 1.04

db10 59.11 61.16 0.08 0.05 1.60 1.57 1.06 1.06

db45 61.26 64.46 0.05 0.02 1.34 1.35 1.07 1.07

sym6 60.56 61.74 0.06 0.04 1.54 1.75 1.07 1.06

coif4 61.25 61.61 0.05 0.04 1.63 1.45 1.12 1.01

bior2.4 61.17 63.65 0.05 0.03 2.15 1.56 1.04 1.07

dmey 58.86 61.53 0.08 0.05 1.55 1.63 1.07 1.08

rbio1.3 62.28 61.21 0.04 0.05 1.33 1.52 1.04 1.05

Average 60.58 62.72 0.06 0.04 1.58 1.52 1.06 1.06



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Table VII describes the performance of multirate BSS LMS by using wavelet packet domain on

denoising. The table shows that the performance of BSS LMS in wavelet packet domain is even better than that

of Var LMS. The average values of PSNR of Var LMS in wavelet packet domain is 60.58dB and 62.72dB with

the two signals while that of BSS LMS is 63.55dB and 62.31dB. The maximum error and ratio of squared norms

are similar with Var LMS and BSS LMS in wavelet packet domain.

Table VII: Denoising with Multirate BSS LMS using Wavelet Packet Domain

Denoising with Multirate BSS LMS using Wavelet Packet Domain



Haar 64.35 59.81 0.02 0.07 2.29 1.82 1.07 1.09

db10 66.74 63.15 0.01 0.03 1.19 1.41 1.08 1.06

db45 61.04 62.29 0.05 0.04 1.40 1.64 1.04 1.09

sym6 62.55 64.95 0.04 0.02 1.34 1.56 1.06 1.03

coif4 59.06 60.45 0.08 0.06 1.94 1.94 1.12 1.07

bior2.4 67.23 62.25 0.01 0.04 1.25 1.36 1.01 1.10

dmey 61.02 62.11 0.05 0.04 1.36 1.53 1.05 1.03

rbio1.3 66.42 63.47 0.01 0.03 1.54 1.32 1.05 1.09

Average 63.55 62.31 0.04 0.04 1.54 1.57 1.06 1.07

The comparison of all these techniques is presented in the table VIII and figures 12 to 15.

Table VIII: Denoising by different techniques

Denoising by different techniques


FS SS FS FS SS FS FS SS

Adaptive Filter 52.68 51.88 0.38 0.47 2.27 3.23 0.95 0.95

DCT &DFT based LMS 53.59 52.33 0.28 0.39 1.94 3.15 0.97 0.91

BSS LMS 53.07 51.98 0.33 0.41 3.08 3.92 0.91 0.88

Var LMS 53.50 51.85 0.29 0.43 3.08 4.03 0.92 0.87

AdFl in WP 54.40 53.99 0.24 0.26 1.52 1.73 1.05 1.07

MultirateVar LMS WP 60.58 62.72 0.06 0.04 1.58 1.52 1.06 1.06

Multirate BSS LMS WP 63.55 62.31 0.04 0.04 1.54 1.57 1.06 1.07



Page 224

Fig. 12 PSNR values with different techniques

Fig. 13 MSE values with different techniques

Fig. 14 Maximum error values with different techniques

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Fig. 15 L2RAT values with different techniques

VII. CONCLUSIONS In this paper the denoising of time division multiplexed version of two signals sampled at different rate

is considered. Traditional adaptive algorithms including DCT and DFT based transform domain LMS, transform domain LMS based on wavelet packets with different wavelets, a new variation of LMS, binary step size LMS

and transform domain version of these two algorithms using wavelet packets with different wavelets are

implemented for the specified denoising. The average PSNR with the traditional adaptive algorithms excluding

DCT and DFT based transform domain LMS is calculated to be 52.28dB, with DCT and DFT based transform

domain LMS 52.96dB, with BSS LMS 52.53dB, with Var LMS 52.67dB, with LMS in wavelet packet domain

54.19dB, Var LMS in wavelet packet domain 61.65dB and with BSS LMS in wavelet packet domain 62.93dB.

From these results one can conclude that the new algorithms devised outperforms the existing techniques.

REFERENCES [1] S.S. Hykin, “Adaptive Filter Theory”, NJ: Prentice – Hall, Englewood Cliffs, 1991.

[2] Dixit S., Nagaria D., “Neural network implementation of Least-Mean-Square adaptive noise cancellation”, IEEE International

Conference on Issues and Challenges in Intelligent Computing Techniques (ICICT), pp. 134 – 139, Feb. 2014

[3] R.H. Kwong and E. W. Johnston, “A variable step size LMS algorithm”, IEEE transactions on signal processing, vol. 40, pp. 1633-

1642, July 1992.

[4] J. C. M. Bermudez and N. J. Bershad, “A nonlinear analytical model for the quantized LMS algorithm: The arbitrary step size case,”

IEEE Transactions on Signal Processing, vol. 44, pp. 1175-1183, May 1996.

[5] Borisagar.K.R, Kulkarni.G.R., “Simulation and Performance Analysis of Adaptive Filter in Real Time Noise over Conventional

Fixed Filter”, IEEE International Conference on Communication Systems and Network Technologies (CSNT), pp. 621 – 624, May

2012.

[6] Haichen Zhao, Shaolu Hu, Linhua Li, Xiaobo Wan, “NLMS adaptive FIR filter design method”, TENCON 2013 - 2013 IEEE

Region 10 Conference, pp. 1 – 5, Oct. 2013.

[7] Koike S, “Performance analysis of adaptive step-size least mean modulus-Newton algorithm for identification of non-stationary

systems”, IEEE 8th International Colloquium on Signal Processing and its Applications (CSPA), pp. 184 – 187, March 2012

[8] Singh A, Bansal B.N., “Analysis of Adaptive LMS Filtering in Contrast to Multirate Filtering”, 4th International Conference on

Emerging Trends in Engineering and Technology (ICETET), pp. 22 – 26, Nov. 2011.

[9] Yuan Feng, ZhihaiZhuo, Shengheng Liu, Xiaokang Cheng, Letao Zhang, Zhipeng Zhang, “A novel adaptive filter based on

FDLMS and modified MFLMS”, IEEE International Conference on Information Science and Technology (ICIST), pp. 1326 – 1329,

March 2013.

[10] Kusljevic M.D., Poljak P.D., “Simultaneous Reactive-Power and Frequency Estimations Using Simple Recursive WLS Algorithm

and Adaptive Filtering”, IEEE Transactions on Instrumentation and Measurement, Vol. 60, Issue 12, pp. 3860 – 3867, Dec. 2011.

[11] Vityazev V.V., Linovich A.Y., “A subband equalizer with the flexible structure of the analysis/synthesis subsystem”, IEEE Region

8 International Conference on Computational Technologies in Electrical and Electronics Engineering (SIBIRCON), pp. 174 – 178,

July 2010

[12] Ramya J.V.S.L, “Alias free sub-band adaptive filtering”, IEEE Conference on Recent Advances in Space Technology Services and

Climate Change (RSTSCC), pp. 119 – 124, Nov. 2010.

[13] Yan Liu, MucongZheng, “Adaptive receiver for multirate CDMA systems on Kalman criterion”, IEEE International Conference on

Computer Science and Service System (CSSS), pp. 2963 – 2965, June 2011

[14] P. S. R. Diniz, E. A. B. da Silva, and S. L. Netto, “Digital Signal Processing: System Analysis and Design”, Cambridge Univer sity

Press, Cambridge, UK, 2002.

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C. Mohan Rao (India) born on 21st April 1975, is pursuing Ph.D in Jawaharlal Nehru

Technological University Anantapur. He received Master of Technology from Pondicherry

Central University in the Year 1999, Bachelor of Technology form S.V University, Tirupati in

the year 1997. He started his carrier as hardware Engineer in Hi-com technologies, during 1999 to 2001, after he worked as Assistant Professor in G. P. R. E. C during 2001 to 2007 and as

Associate Professor in N. B. K. R. I. S. T. from 2007 to till date. He is a member of IETE and

ISTE.

Dr. B. Stephen Charles (India) born on the 9th of August 1965. He received Ph.d degree in

Electronics & Communication Engineering from Jawaharlal Nehru Technological University,

Hyderabad in 2001. His area of interest is Digital signal Processing. He received his B.Tech

degree in Electronics and Communication Engineering from Nagarjuna University, India in

1986. He started his carrier as Assistant professor in Karunya institute of technology during

1989 to 1993, later joined as Associate Professor in K. S. R. M. College of Engg. During 1993 to 2001 after that he worked as Principal of St. John's College of Engineering & Technology

during 2001 to 2007 and now he is the Secretary, Correspondent and Principal in Stanley

Stephen College of Engineering & Technology, Kurnool. He has 24 years of teaching and

research experience. He published more than 40 research papers in national and international

journals and more than 30 research papers in national and international conferences. He is a

member of Institute of Engineers and ISTE.

Dr. M.N. Giri Prasad received his B.Tech degree from J.N.T University College of Engineering,

Anantapur, Andhrapradesh, India in 1982. M.Tech degree from Sri Venkateshwara University,

Tirupati, Andhra Pradesh, India in 1994 and Ph.D degree from J.N.T. University, Hyderabad, Andhra Pradesh, Indian in 2003. Presently he is working as a Professor in the Department of

Electronics and Communication at J.N.T University College of Engineering Anantapur,

Andhrapradesh, India. He has more than 25 years of teaching and research experience. He has

published more than 50 papers in national and international journals and more than 30 research

papers in national and international conferences. His research areas are Wireless

Communications and Biomedical instrumentation, digital signal processing, VHDL coding and

evolutionary computing. He is a member of ISTE, IE & NAFEN.

denoising of tdm signal using novel time domain and transform domain multirate adaptive algorithms

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